Usually, exponents of numbers appear in algebraic formulas and equations. We discuss both positive exponents and negative exponents, and even complex exponents. We show you how to handle such numbers perfectly.

What are exponents of numbers?

An exponent refers to the number of times a number is multiplied by itself. Let us for example take as number $5$. If $3$ is the exponent of $5$, then we obtain another number $5\times 5\times 5$, the number five is repeated $3$ times if $3$ is the exponent of $5$. In this case, we write $5^3$. Here $5$ is called the base, $3$ is the exponent and $5^3$ is the power.


Let us now give a more general formula for power numbers. Let $y$ be a real number and $n$ a natural number. We write \begin{align*} y^n=\underset{n \; {\rm times}}{\underbrace{y\times y\cdots\times y}}\end{align*}

Some exercises 

Exercise: let $a$ and $b$ be two numbers. Prove that \begin{align*}(ab)^2=a^2b^2.\end{align*}

Solution: We know that $ab=ba$. Then by definition \begin{align*}(ab)^2=(ab)(ab)=a(ba)b=a(ab)b=(aa)(bb)=a^2b^2.\end{align*}More generally, if  $n$ is a natural number, then\begin{align*}

Exercise: Prove that\begin{align*} \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}.\end{align*}
Solution: We will apply the above formula of exponent of product numbers. In fact, we set $$c=\frac{a}{b}.$$
Then $bc=a$. Now we take the exponent on both sides of this equality, we get $(bc)^n=a^n$. But we know that $(bc)^n=b^nc^n$. Thus $b^nc^n=a^n.$ By dividing both sides by $b^n,$ we obtain \begin{align*}
\left(\frac{a}{b}\right)^n=c^n=\frac{a^n}{b^n}.\end{align*} Let $n$ and $m$ be two integers and $y$ be a real number. Then \begin{align*} y^{n+m}=y^n y^m\end{align*}

Application: For integers numbers $n$ and $m,$ prove that\begin{align*}  \left(y^{n}\right)^m=y^{nm}.\end{align*} In fact, by definition of exponents, we have\begin{align*} \left( y^{n}\right)^m&=y^n\cdot  y^n\cdots  y^n\;(m\; \times)\cr & = y^{n+n+\cdots+n}.\end{align*}But the number $n$ is repeated  $m$ times in the sum $n+n+\cdots+n$. Thus  $n+n+\cdots+n=mn$. The result then follows.

Example: Compare the numbers $8^4$ and $2^{12}$.\begin{align*}2^{12}= 2^{3\times 4}=(2^3)^4=8^4.\end{align*}

Exercise: Find the number $x$ such that $x^5=2^{15}$.

Solution: Let us first recall the following rule: $ a^n = b^n $ implies that $ a = b $. We will apply this property to answer the exercise. In fact, the idea is to rewrite $2^{15}$ as an exponent of $5$. Remark that $15=3\times 5$. Then\begin{align*}x^5=2^{15}=2^{3\times 5}=(2^3)^5=8^5.\end{align*}It follows that $x=8$.

 Negative  exponents

In the previous section, we saw the definition and properties of positive exponents of numbers. In this section, we will discuss the case of negative exponents of numbers. By definition, if $a$ is a real number and $n$ is an integer, then \begin{align*}a^{-n}:=\frac{1}{a^n}.\end{align*}For $n=1,$ we have \begin{align*}a^{-1}=\frac{1}{a}.\end{align*}This shows that $$(a^{-1})^n=a^{-n}.$$On the other hand, if $n$ and $m$ are integers such that $n\ge m$, then we can write $a^n=a^{(n-m)+m}=a^{n-m}a^m$. By dividing the two sides of this equation by $a^m$, we get\begin{align*}a^{n-m}=a^n\times \frac{1}{a^m}=a^n a^{-m}.\end{align*}From this formula, we conclude that for any relative numbers, positive or negative, $n$ and $m$, we have\begin{align*} y^{n+m}=y^n  y^m.\end{align*}

Exercise: Simplify the expression\begin{align*}A(n)=\frac{2^n+1}{1+2^{-n}}.\end{align*}

Solution: We factor the numerator of the fraction by $2^n,$ we get\begin{align*}A(n)=2^n\frac{1+2^{-n}}{1+2^{-n}}=2^n.\end{align*}

You can also consult the page on the powers of numbers where you will find some exercises with detailed solutions.